President Barack Obama will likely soon have three top staff positions to fill, at a time when he is under mounting pressure from the public and Democratic lawmakers to prove he can manage more adeptly and restore confidence in his signature health-care program.

The White House said over the weekend that it would create a new, permanent position overseeing HealthCare.gov, the troubled website that has threatened the success of the 2010 health law and been one of the most serious challenges facing the Obama presidency....